Mαθemaτιcal Formulation

  • Shape: A Shape is modeled as a metric space

    • Abstract & Euclidean Riemannian manifolds
    • abstract metric graphs
    • embedded metric graph (𝒢⊂ℝN\mathcal G\subset\mathbb R^N)

  • Sample: A finite S⊂ℝNS\subset\mathbb R^N with small Hausdorff distance dH(𝒢,S)d_{\mathrm H}(\mathcal G, S)

  • Goal: Infer the topology of 𝒢\mathcal G from SS.

    • construct a topological space 𝒢̂\hat{\mathcal G} (e.g. Vietoris–Rips) from SS so that
      • 𝒢̂\hat{\mathcal G} is homotopy equivalent to 𝒢\mathcal G (topological reconstruction)
      • 𝒢̂⊂ℝN\hat{\mathcal G}\subset\mathbb R^N and dH(𝒢̂,S)d_{\mathrm H}(\hat{\mathcal G}, S) small (geometric reconstruction)

Definition

  • a metric space (X,dX)(X,d_X)

  • a scale β>0\beta>0

  • ℛβ(X)\mathcal{R}_\beta(X) is an abstract simplicial complex

    • XX is the vertex set
    • each subset A⊂XA\subset X of (k+1)(k+1) points with diameter less than β\beta is a kk-simplex.

  • Persistent homology considers all possible scales β\beta to make reasonable (only) homological inference

  • My goal is to provide a [window] of scales where homotopy properties are guaranteed.

Why Does Vietoris–Rips Make Sense?

  • Computationally efficient
  • Data dimension agnostic—unlike the Čech complex or α\alpha-complex

Hausmann (1995)

For any closed Riemannian manifold XX and 0<β<ρ(X)0<\beta<\rho(X), the Vietoris–Rips complex ℛβ(X)\mathcal{R}_\beta(X) is homotopy equivalent to XX.

Latschev (2001)

Every closed Riemannian manifold XX has an ϵ0>0\epsilon_0>0 such that for any 0<β≤ϵ00<\beta\leq\epsilon_0 there exists some δ>0\delta>0 so that for any sample SS: dGH(S,X)≤δ⟹ℛβ(S)≃X. d_{GH}(S,X)\leq\delta\implies \mathcal R_\beta(S)\simeq X.

Quantitative Latschev’s Theorems

Vietoris–Rips Complexes in Limits

  • (X,dX)(X, d_X) metric space and indexed set: 𝕊≔{S⊂X:|S|<∞}\mathbb{S}\coloneqq\{S\subset X\ : \lvert S\rvert<\infty\} under set inclusion
  • For any scale β≥0\beta\geq0, we have the direct system of groups {πm(ℛβ(S))→πm(ℛβ(T))∣S,T∈𝕊,S⊂T}\left\{\pi_m(\mathcal{R}_\beta(S))\to \pi_m(\mathcal{R}_\beta(T))\mid S,T\in\mathbb{S}, S\subset T\right\}

Hausmann in Limits (Kawamura, Majhi, and Mitra 2026)

The direct limit: πm(ℛβ(𝕊))≔limS∈𝕊πm(ℛβ(S))≅πm(ℛβ(X)). \pi_m(\mathcal{R}_\beta(\mathbb{S}))\coloneq\lim_{S\in\mathbb{S}}\pi_m(\mathcal{R}_\beta(S))\cong \pi_m(\mathcal{R}_\beta(X)).

Latschev in Limits (Kawamura, Majhi, and Mitra 2026)

Let D≔{pk∣k=1,2,…}D\coloneqq\{ p_{k}\mid k=1,2,\ldots\} be a countable dense subset of a separable space XX and let Sk≔{pi∣i=1,…,k}S_{k}\coloneq \{ p_{i}\mid i=1,\ldots, k\}. Then, limkπm(ℛβ(Sk))≅πm(ℛβ(X)).\lim_{k}\pi_m(\mathcal{R}_\beta(S_{k})) \cong \pi_m(\mathcal{R}_\beta(X)).

Challenges

  • Unknown Shape: 𝒢⊂ℝN\mathcal{G}\subset\mathbb R^N with geodesic metric d𝒢d_\mathcal{G}
  • Sample: S⊂ℝNS\subset\mathbb{R}^N with Euclidean metric ∥•∥\|\bullet\|
  • Density: dH(S,𝒢)d_H(S,\mathcal{G}) small or can be controlled
  • Q: Find a [range] of β\beta so that ℛβ(S)\mathcal{R}_\beta(S) is homotopy equivalent to 𝒢\mathcal{G}.

Bad News :-(

A single Vietoris–Rips complex fails to be topologically faithful, no matter the sample density

Remedy 1: ε\varepsilon-path Metric

  1. Fix a positive ε\varepsilon: proportional to dH(S,𝒢)d_H(S, \mathcal G);

  2. Compute ε\varepsilon-neighborhood graph on SS: ℛε(1)(S)\mathcal R^{(1)}_\varepsilon(S);

  3. Define dSε(a,b)d_{S}^\varepsilon(a,b) to be the shortest path metric on ℛε(1)(S)\mathcal R^{(1)}_\varepsilon(S);

  4. Denote by ℛβε(S)\mathcal{R}_\beta^\varepsilon(S) the Vietoris–Rips complex of SS under dSεd^\varepsilon_{S}.

ε\varepsilon-path Metric on SS

Quasi-isometry (Majhi 2023)

For a dense enough sample SS of 𝒢\mathcal G, dSεd^\varepsilon_{S} is quasi-isometric to the length metric on 𝒢\mathcal G.

Remedy 2: Large-scale Distortion

  • A finite sample around a corner does not see the corner
  • Global distortion: δ(𝒢)=supa≠bd𝒢L(a,b)∥a−b∥\delta(\mathcal G)=\sup_{a\neq b}\frac{d^L_{\mathcal G}(a,b)}{\|a-b\|}
  • Large-scale distortion: δRε(𝒢)=supdL(a,b)≥Rd𝒢L(a,b)d𝒢εL(a,b)\delta^\varepsilon_R(\mathcal G)=\sup_{d^L(a,b)\geq R}\frac{d^L_{\mathcal G}(a,b)}{d^L_{\mathcal G^\varepsilon}(a,b)}

Limit (Komendarczyk, Majhi, and Tran 2024)

For any R>0R>0, δRε(𝒢)→1\delta^\varepsilon_R(\mathcal G)\to1 as ε→0\varepsilon\to0, provided 𝒢\mathcal G is compact.

Topological Reconstruction

Metric Graph Reconstruction (Komendarczyk, Majhi, and Mitra 2025)

Let 𝒢⊂ℝN\mathcal G \subset \mathbb{R}^N be a compact, connected metric graph.
Fix any ξ∈(0,14)\xi\in\left(0,\frac{1}{4}\right). For any positive β<ℓ(𝒢)4\beta<\frac{\ell(\mathcal G)}{4}, choose a positive ε≤β3\varepsilon\leq\frac{\beta}{3} such that δβε(𝒢)≤1+2ξ1+ξ\delta^{\varepsilon}_{\beta}(\mathcal G)\leq\frac{1+2\xi}{1+\xi}.

If S⊂ℝNS\subset \mathbb R^N is such that dH(𝒢,S)<12ξεd_H(\mathcal G,S)<\tfrac{1}{2}\xi\varepsilon, then we have a homotopy equivalence ℛβε(S)≃𝒢\mathcal R^\varepsilon_\beta(S)\simeq \mathcal G.

Fixing ξ=1/6\xi=1/6, we get a simpler but weaker statement.

Let 𝒢⊂ℝN\mathcal G \subset \mathbb{R}^N be a compact, connected metric graph.
For any positive β<ℓ(𝒢)/4\beta<\ell(\mathcal G)/4, choose a positive ε≤β/3\varepsilon\leq\beta/3 such that δβε(𝒢)≤8/7\delta^{\varepsilon}_{\beta}(\mathcal G)\leq8/7. If S⊂ℝNS\subset \mathbb R^N and dH(𝒢,S)<ε/12d_H(\mathcal G,S)<\varepsilon/12, then we have a homotopy equivalence ℛβε(S)≃𝒢\mathcal R^\varepsilon_\beta(S)\simeq\mathcal G.

Vietoris–Rips Shadow

  • Geometric reconstruction: entails constructing 𝒢̂⊂ℝN\hat{\mathcal G}\subset\mathbb R^N from samples so that
    • 𝒢̂≃𝒢\hat{\mathcal G}\simeq \mathcal G & dH(𝒢̂,𝒢)d_H(\hat{\mathcal G}, \mathcal G) is small
  • Vietoris–Rips complexes are abstract, hence contain only topological information
  • A good candidate for 𝒢̂\hat{\mathcal G} is the shadow of a topologically-faithful Vietoris–Rips.

𝒦\mathcal K

  • Shadow: For simplicial complex 𝒦\mathcal K with vertices from ℝN\mathbb R^N is the union of the (Euclidean) convex hulls of simpices σ∈𝒦\sigma\in\mathcal K

  • Notorious for being topologically unfaithful

    • Chambers et al. (2010); Adamaszek, Frick, and Vakili (2017)

𝒮(𝒦)\mathcal S(\mathcal K)

Vietoris–Rips Shadow in Limits

  • X⊂ℝNX\subset\mathbb R^N Euclidean metric space (Submanifolds, graphs; more generally ANR)
  • Denote πm(𝒮β(𝕊))≔limS∈𝕊πm(S(ℛβ(S)))\pi_{m}(\mathcal{S}_\beta(\mathbb{S})) \coloneq \lim_{S\in \mathbb{S}} \pi_{m}(S(\mathcal{R}_\beta (S)))
  • We obtain an inverse system: {πm(𝒮γ(𝕊))→πm(𝒮β(𝕊))∣0<γ<β} \left\{ \pi_{m}(\mathcal{S}_{\gamma}(\mathbb{S})) \to \pi_{m}(\mathcal{S}_{\beta}(\mathbb{S})) \mid 0 < \gamma < \beta \right\}
  • The inverse limit group: limβπm(𝒮β(𝕊)) \lim_{\beta} \pi_{m}(\mathcal{S}_{\beta}(\mathbb{S}))

Hausmann in Shadow Limits (Kawamura, Majhi, and Mitra 2026)

limβπm(𝒮β(𝕊))≅πm(X). \lim_{\beta} \pi_m(\mathcal{S}_\beta(\mathbb S)) \cong \pi_m(X).

Graph Reconstruction via Shadow

Geometric Reconstruction (Komendarczyk, Majhi, and Mitra 2025)

Let 𝒢⊂ℝ2\mathcal G \subset \mathbb{R}^2 a graph. Fix any ξ∈(0,1−Θ6)\xi\in\left(0,\frac{1-\Theta}{6}\right). For any positive β<min{Δ(𝒢),ℓ(𝒢)12}\beta<\min\left\{\Delta(\mathcal G),\frac{\ell(\mathcal G)}{12}\right\}, choose a positive ε≤(1−Θ)(1−Θ−6ξ)12β\varepsilon\leq\frac{(1-\Theta)(1-\Theta-6\xi)}{12}\beta such that δβε(𝒢)≤1+2ξ1+ξ\delta^{\varepsilon}_{\beta}(\mathcal G)\leq\frac{1+2\xi}{1+\xi}.

If S⊂ℝ2S\subset \mathbb R^2 and dH(𝒢,S)<12ξεd_H(\mathcal G, S)<\tfrac{1}{2}\xi\varepsilon, then the shadow 𝒮(ℛβε(S))\mathcal{S}(\mathcal R_\beta^\varepsilon(S)) is homotopy equivalent to 𝒢\mathcal G. Moreover, dH(𝒮(ℛβε(S)),𝒢)<(β+12ξε)d_H(\mathcal S(\mathcal R_\beta^\varepsilon(S)),\mathcal G)<\left(\beta+\frac{1}{2}\xi\varepsilon\right).

  • Θ∈(0,1)\Theta\in(0,1): depends on the angles between tangents of edges at the graph vertices
  • Δ(G)\Delta(G): Shadow radius positive number for graphs with smooth edges
  • 𝒢\mathcal G is planar

Future Directions

  • The general case: 𝒢⊂ℝN\mathcal G\subset\mathbb R^N

    • in preparation with Kazuhiro Kawamura & Atish Mitra

  • Use discrete Morse theory to perform organized collapses of higher dimensional simplices of shadow for homeomorphic reconstruction

  • Submanifold reconstruction via Vietoris–Rips shadow